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Background

Please note that I am duplicating the question on scicomp. I have already asked this in math.

I am trying to come up with a scheme in Polar Co-ordinates for the following PDE:

PDE I am trying to solve $$ u_t - (u_{rr} + \frac{1}{r} * u_r + \frac{1}{\theta} * u_{\theta\theta} + bu) = f(r,\theta, t)$$

I am extending the ideas from the book [1] which is already given for the PDE:

PDE in [1]

$$u_{rr} + \frac{1}{r} * u_r + \frac{1}{\theta} * u_{\theta\theta} = f(r,\theta, t)$$

The book discusses a particular scheme in pages 333-334 to solve the above PDE. I am listing the scheme below for convenience and its notation.

Scheme proposed for the above PDE in [1]

$$\frac{1}{r_i} (r_{i+1/2} \frac{u_{i+1j} - u_{ij}}{\Delta r} - r_{i-1/2}\frac{u_{ij} - u_{i-1j}}{\Delta r}) \frac{1}{\Delta r} + \frac{1}{r_i^2} \frac{u_{ij+1} - 2u_{ij} + u_{ij-1}}{\Delta \theta^2} = f_{ij}$$

where $u_{ij}$ and $f_{ij}$ are the grid functions at $(r_i, \theta_j) = (i\Delta r, j\Delta \theta)$

Further, to avoid difficulties arising at the origin because of the $\frac{1}{r}$ terms in the scheme, the book goes on to integrate the PDE over a small disc (D) of radius $\epsilon$ near the origin $r=0$, applies the Gauss-Divergence Theorem to get the following result:

$$\iint_D f \;r\;dr\;d\theta = \iint_D (u_{rr} + \frac{1}{r} * u_r + \frac{1}{\theta} * u_{\theta\theta}) \; \;r\;dr\;d\theta = \iint_D \nabla. (\nabla u) \; \;r\;dr\;d\theta = \int_{0}^{2\pi} \frac{\partial u}{\partial r} \;\epsilon\;d\theta$$

These are later approximated as follows:

$$f(0) * \epsilon^2 * \pi = \sum_{j=1}^{J} \frac{u_{1j}-u_0}{\Delta r} \epsilon\; \Delta \theta$$

From which it solves for $u_0$.

Note: $J\Delta \theta = 2 \pi$ and $I\Delta r = R$ where R: radius of the domain.

Question

i) How do I extend the idea for the terms $u_t$, $bu$ terms above when computing an approximation for $u_0$ for my PDE?

I propose to treat $bu$ term where $b$ is a constant the same way as $f$ is treated in the approximation above. But it is unclear to me as to what I need to do for $u_t$.

Any help is greatly appreciated. Even pointers to read a particular book/topic.

Thanks!

References: [1] Elliptic Partial Differential Equations and Difference Schemes Finite Difference Schemes and Partial Differential Equations, Second Edition. 2004, 311-338

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  • $\begingroup$ For $u_t$, you will typically apply a timestepping scheme and approximate this with a forward/backwards difference or some fancier multistep scheme. This means that you'll need to use your solution at the current time to bootstrap your solution at the next time. $\endgroup$ – Jesse Chan Dec 12 '14 at 5:40
  • $\begingroup$ Please understand that cross posting the exact same question on multiple SE sites is highly discouraged. You need to keep only one copy of the question on the most appropriate site and delete the other, or modify them to be distinct from each other. $\endgroup$ – Paul Dec 12 '14 at 6:41
  • $\begingroup$ Could you add a link to the question on MSE? It is good practice here to cross-link double posts, so that readers can see the (future) answers and the discussion on both sites. $\endgroup$ – Federico Poloni Dec 12 '14 at 16:25
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    $\begingroup$ Your questions is really just "how to discretize a time-dependent PDE in the time dimension." Your PDE is not elliptic; look at any book that discusses discretization of parabolic PDEs. $\endgroup$ – David Ketcheson Dec 13 '14 at 4:12
  • $\begingroup$ @hardmath, I have not done Finite Elements. I am not sure that I can do it with FE at the moment. $\endgroup$ – mod0 Dec 13 '14 at 15:30

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